A simple and explicit algebraic expression for the Rayleigh wave velocity

Nkemzi, Daniel

doi:10.1016/j.mechrescom.2007.10.005

Cited by 13 publications

(14 citation statements)

References 11 publications

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“…It appears that the first paper that published explicit expressions for the Rayleigh wave speed for the full range of elastic material properties was by Rahman and Barber [2]. Since that time, a number of authors have sought to develop alternative analytical expressions for the Rayleigh wave speed [3][4][5][6][7][8][9][10][11]. It is noted that the solutions provided cannot be used indiscriminately, as care is required on occasions to choose the correct root to ensure a smooth and continuous estimate of the Rayleigh wave speed [3,5,7].…”

Section: ‡ Introductionmentioning

confidence: 99%

“…Complete analytical derivations were provided by [2,4,7,9,11]. Others have used computer algebra to assist in their derivations [3,5,6].…”

Section: ‡ Introductionmentioning

confidence: 99%

“…Others have used computer algebra to assist in their derivations [3,5,6]. The original approach given in [4] contains unspecified typographical errors [5,9]. Indeed, the recent solution given in [9] appears to have been derived earlier and independently but with typographical errors [6].…”

Section: ‡ Introductionmentioning

confidence: 99%

“…The original approach given in [4] contains unspecified typographical errors [5,9]. Indeed, the recent solution given in [9] appears to have been derived earlier and independently but with typographical errors [6]. It has also been shown to be identical to the solution provided in [5].…”

Section: ‡ Introductionmentioning

confidence: 99%

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Use of Padé Approximants to Estimate the Rayleigh Wave Speed

Spathis¹

2015

TMJ

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There exists a range of explicit and approximate solutions to the cubic polynomial Rayleigh equation for the speed of surface waves across an elastic half-space. This article presents an alternative approach that uses Padé approximants to estimate the Rayleigh wave speed with five different approximations derived for two expansions about different points. Maximum relative absolute errors of between 0.34% and 0.00011% occur for the full range of the Poisson ratio from-1 to 0.5. Even smaller errors occur when the Poisson ratio is restricted within a range of 0 to 0.5. For higher-order approximants, the derived expressions for the Rayleigh wave speed are more accurate than previously published solutions, but incur a slight cost in extra arithmetic operations, depending on the desired accuracy.

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Section: ‡ Introductionmentioning

confidence: 99%

“…Complete analytical derivations were provided by [2,4,7,9,11]. Others have used computer algebra to assist in their derivations [3,5,6].…”

Section: ‡ Introductionmentioning

confidence: 99%

Section: ‡ Introductionmentioning

confidence: 99%

Section: ‡ Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Use of Padé Approximants to Estimate the Rayleigh Wave Speed

Spathis¹

2015

TMJ

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“…Although the Rayleigh wave was predicted more than 125 years ago by Lord Rayleigh (1885), explicit solutions for the isotropic Rayleigh velocity as a function of the bulk velocities, as opposed to the secular equation, were only recently derived (Rahman and Barber, 1995;Mechkour, 2002). This solution was later analyzed by several authors to develop an even simpler approximate expression for the velocity in isotropic and anisotropic media using various mathematical techniques (Vinh and Ogden, 2004b;Li, 2006;Rahman and Michelitsch, 2006;Nkemzi, 2008). Among the explicit solutions derived for the Rayleigh-wave velocity, Vinh and Ogden (2004a) obtained a simple expression for the Rayleigh velocity in orthotropic media as a function of the elastic constants (see equation 3.28 in Vinh and Ogden, 2004a).…”

Section: Introductionmentioning

confidence: 99%

Measurements of elastic constants in anisotropic media

2014

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Simulation of elastic-wave propagation in rock requires knowledge of the elastic constants of the medium. The number of elastic constants required to describe a rock depends on the symmetry class. For example, isotropic symmetry requires only two elastic constants, whereas transversely isotropic symmetry requires five unique elastic constants. The off-diagonal elastic constant depends on a wave velocity measured along a nonsymmetry axis. The most difficult barrier when measuring these elastic constants is the ambiguity between the phase and group velocity in experimental measurements. Several methods to eliminate this difficulty have been previously proposed, but they typically require several samples, difficult machining, or complicated computational analysis. Another approach is to use the surface (Rayleigh) wave velocity to obtain the off-diagonal elastic constant. Rayleigh waves propagated along symmetry axes have phase and group velocities that are equal for materials with no frequency dispersion, thereby eliminating the ambiguity. Using a theoretical secular equation that relates the Rayleigh velocity to the elastic constants enable determination of the offdiagonal elastic constant. Laboratory measurements of the elastic constants in isotropic and anisotropic materials were made using ultrasonic transducers (central frequency of 1 MHz) for the Rayleigh-wave method and a wavefront-imaging method. The two methods indicated agreement within 1% and 3% for isotropic and transversely isotropic samples, respectively, demonstrating the ability of the Rayleigh-wave method to measure the off-diagonal elastic constant. The surface-wave approach eliminates the need for multiple samples, expensive computational calculations, and most importantly, it removes the ambiguity between the phase and group velocity in the measured data for materials with no frequency dispersion because all measurements are made along symmetry axes.

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Interaction of Underwater Blasts and Submerged Structures

Abrate

2012

Dynamic Failure of Composite and Sandwich Structures

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A simple and explicit algebraic expression for the Rayleigh wave velocity

Cited by 13 publications

References 11 publications

Use of Padé Approximants to Estimate the Rayleigh Wave Speed

Use of Padé Approximants to Estimate the Rayleigh Wave Speed

Measurements of elastic constants in anisotropic media

Interaction of Underwater Blasts and Submerged Structures

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